Download Chapter 18. Molecular interactions

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Atkins / Paula
《 Physical Chemistry, 8th Edition 》
Chapter 18.
Molecular interactions
In this chapter we examine molecular interactions in
gases and liquids and interpret them in terms of
electric properties of molecules, such as electric dipole
moments and polarizabilities.
在本章我們將檢視氣體和液體分子的交互作用並且根據分子電
性質解釋他們的電偶極矩和極化率。
Chap 18. Molecular interactions
Chapter 18. Molecular interactions
Table of Contents 目錄內容
Electric properties of molecules 分子的靜電性質
18.1 Electric dipole moments 電偶極矩
18.2 Polarizabilities 極化性
18.3 Relative permittivities 相對電容率(介電常數)
Interactions between molecules 分子間的作用力
18.4 Interactions between dipoles 偶極矩間的作用力
18.5 Repulsive and total interactions 斥力與分子總作用力
Gases and liquids 氣體與液體中的分子作用
18.6 Molecular interactions in gases 氣體中的分子作用
18.7 The liquid-vapor interface 氣液體介面中的分子作用
18.8 Condensation 凝結作用
Chap 18. Molecular interactions
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• All these electric properties of molecules, such as
electric dipole moments and polarizabilities, reflect
the degree to which the nuclei of atoms exert control
over the electrons in molecule, either by causing
electrons to accumulate in particular regions, or by
permitting them to respond more or less strongly to
the effects of the external electric fields.
• 分子所有這些電性質，例如電偶極矩和極化率，反映出分子
中原子核對電子的控制，造成電子在特定區域的累積程度，
或是外在電場作用下電子容許有多少的反應。
Chap 18. Molecular interactions
Electric properties of molecules
• Many of electric properties of molecules can be traced to
分子的許多電性的成因可以被追蹤到有如下:
 the competing influences of nuclei with different
charges, or 原子核對不同電荷影響力的競爭,或
 the competition between the control exercised by a
nucleus and the influence of an externally applied
field. 原子核的控制力與外加電場影響力之間的競爭。
• The former competition may result in an electric dipole
moment. The latter may result in properties such as
refractive index and optical activity. 前者競爭可導致偶極矩
。後者可導致例如折射率和光學活性等物性。
Chap 18. Molecular interactions
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18.1 Electric dipole moments
• An electric dipole moment µ consists of two equal
and opposite charges, q , separated by a distance R.
• The moment of dipole, the dipole moment µ, is a
vector of magnitude µ =q R, that points from the
negative charge (-) to the positive charge (+).
• Dipole moment are normally reported in the cgs
unit Debye but their SI unit is coulomb-meter (C m).
• The original definition of one Debye is the dipole
moment of two equal and opposite charges of
magnitude 1 e.s.u. separated by 1 Å.
1 D =3.33564x10-30 C m
Chap 18. Molecular interactions
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6
• In elementary chemistry,
an electric dipole moment
is represented by the
arrow ↛ added to the
Lewis structure for the
molecule, with the +
marking the positive end.
Note that the direction of
the arrow is opposite to
that of μ.
• 在普通化學課本，由箭頭↛代
表電偶極矩，附加於分子
Lewis結構，正端標記+。 注
意箭頭方向與 μ 定義不同。
Chap 18. Molecular interactions
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• The dipole moment of a pair of charges +e and -e
separated by 100 pm is 1.6 x x10-19 C m,
corresponding to 4.8 D.
一對偶極矩電荷 若是由+e 和 -e分離的距離為100 pm為
1.6 x10-19 C m，對應到4.8 D。
• Dipole moments of small molecules are typically about
1 D.
典型的小分子的偶極矩是大約1 D。
Chap 18. Molecular interactions
Polar molecular
• Polar molecular (極性分子)
Molecules with permanent non-zero electric dipole moments
are classified as polar molecules. The permanent dipole
moment arise from differences in electronegativity which
results in atoms possessing partial charges or imbalance in
the distribution of electric charge or other features of
bonding. (e.g. CO molecule)
分子有永久的非零電偶極矩則被分類為極性分子。當原子的陰
電性差別，導致部份電荷不平衡；或從其他鍵結特性，結果均
可出現永久偶極矩的分子。 (例如。 CO分子)
• A dipole moment exists in a polar molecule can also be
modified by an applied field.
一個極性分子存在的偶極矩，可能被外加電場修飾。
Chap 18. Molecular interactions
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Non-polar molecular
9
• Non-polar molecule (非極性分子)
• Nonpolar molecules acquire an induced dipole moment
in an electric field on account of the distortion the field
causes in their electronic distributions and nuclear
positions; (一個非極性分子在一個電場獲取了感應電偶極矩，
是由於電場在片刻造成的他們的電子分佈和核位置的畸變。)
• However, this induced moment is only temporary, and
disappears as soon as the perturbing field is removed.
Polar molecules also have their existing dipole moments
temporarily modified by an applied field. (然而，這感應電
偶極矩只是臨時的，並且當去除干擾的外加電場時感應電偶極
矩即消失。極性分子也可感應外加電場，而暫時地修改他們現
有的永久偶極矩。)
Chap 18. Molecular interactions
Stark effect
• Stark effect is used to measure the electric dipole
moments of molecule for which a rotational spectrum
can be observed. In many cases microwave
spectroscopy cannot be used because the sample is
not volatile, decomposes on vaporization, or consists
of molecules that are so complex that their rotational
spectra cannot be interpreted. (當分子的旋轉光譜可以
被觀察時，Stark 效應用於測量分子的電偶極矩。許多情況
下因為樣品不揮發，或在汽化時分解，或分子包括很複雜的
結構，使得旋轉光譜無法解釋。則不能使用微波光譜。)
• In such cases the dipole moment may be obtained by
measurement on a liquid or solid bulk sample using
others method. (此時需以其他方法量測液態或固態樣品.)
Chap 18. Molecular interactions
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• All heteronuclear diatomic molecules are polar, and
typical values of μ include 1.08 D for HCl and 0.42 D
for HI (Table 18.1). (所有異核雙原子分子皆有極性，並且μ
的典型值例如(表18.1): HCl為1.08 D ; HI為0.42 D)
• Molecular symmetry is of the greatest importance in
deciding whether a polyatomic molecule is polar or
not. (在決定一個多原子分子是否有極性時，分子對稱才是
最重要的因素。)
• Indeed, molecular symmetry is more important than
the question of whether or not the atoms in the
molecule belong to the same element. (分子對稱的重要
性的確超過在分子中的原子是否屬於同一個元素的問題。)
Chap 18. Molecular interactions
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Chap 18. Molecular interactions
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• Homonuclear polyatomic molecules may be polar if
they have low symmetry and the atoms are in
inequivalent positions. (同核多原子分子如果他們有低的
對稱性且原子是在不對等的位置，則可能為極性的。)
• For instance, the angular molecule ozone, O3 (2), is
homonuclear; however, it is polar because the central
O atom is different from the outer two (it is bonded to
two atoms, they are bonded only to one); moreover,
the dipole moments associated with each bond make
an angle to each other and do not cancel. (舉例來說，
角型的臭氧分子，臭氧O3（ 2 ）是同核的但也是極性的，因
為中央的O原子是由不同的外二（它是鍵結兩個原子的Ｏ，
其他的是只有鍵結一個原子的Ｏ）; 此外，每一個鍵結的偶
極矩，形成的角度，彼此不抵消。)
Chap 18. Molecular interactions
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Chap 18. Molecular interactions
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Chap 18. Molecular interactions
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• Heteronuclear polyatomic molecules may be nonpolar
if they have high symmetry, because individual bond
dipoles may then cancel. (異核多原子分子如果有高對稱
性，可以是非極性的。因為個別鍵結的偶極矩，可能會相互
抵消。)
• The heteronuclear linear triatomic molecule CO2 for
example, is nonpolar because, although there are
partial charges on all three atoms, the dipole moment
associated with the OC bond points in the opposite
direction to the dipole moment associated with the CO
bond, and the two cancel (3). (舉例來說，線性異核三原
子分子的CO2為非極性，因為，雖然在所有三個原子上 有
部分電荷，但在OC鍵結上的偶極矩指向CO鍵結上偶極矩的
相反方向，使偶極矩兩者間相抵消（見結構圖3）。)
Chap 18. Molecular interactions
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Chap 18. Molecular interactions
18
Chap 18. Molecular interactions
Dipole moment of a polyatomic molecule
• To a first approximation, it is possible to resolve the
dipole moment of a polyatomic molecule into
contributions from various groups of atoms in the
molecule and the directions in which these individual
contributions lie (Fig. 18.1).
• Thus, 1,4-dichlorobenzene is nonpolar by symmetry
on account of the cancellation of two equal but
opposing C-C1 moments(exactly as in carbon
dioxide).
• 1,2-Dichlorobenzen, however, has a dipole moment
which is approximately the resultant of two
chlorobenzene dipole moments arranged at 60 to
each other,
Chap 18. Molecular interactions
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18.1 The resultant
dipole moments
(grey) of the
dichlorobenzen
e isomers (b to
d) can be
obtained
approximately
by vectorial
addition of two
chlorobenzene
dipole moments
(1.57 D).
Chap 18. Molecular interactions
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18.2 The resultant
dipole moments
(grey) of the
dichlorobenzen
e isomers (b to
d) can be
obtained
approximately
by vectorial
addition of two
chlorobenzene
dipole moments
(1.57 D).
Chap 18. Molecular interactions
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Chap 18. Molecular interactions
23
• This technique of ‘vector addition’ can be applied with
fair success to other series of related molecules, and
the resultant µ res of two dipole moments µ1 and µ2
that make an angle θ to each other (4) is
approximately
µ res 2 = µ12 + µ22 + 2µ1µ2 cos θ
• When the two dipole moments have the same
magnitude (as in the dichlorobenzenes), this equation
simplifies to
µ res 2 = 2µ12 + 2µ12cos θ
= 2µ12 (1+cos θ) = 4µ12 cos2(θ/2)
µ res
= 2µ1 cos (θ/2)
Chap 18. Molecular interactions
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Chap 18. Molecular interactions
25
• Self-test 18.1.
• Estimate the ratio of the electric dipole moments of
ortho (1,2-) and meta (1,3-) di-substituted benzenes.
[µ (ortho)/ µ (meta) =1.7]
Ans: Since µ = 2µ1 cos (θ/2) ;
µ (ortho)/µ (meta) = cos (60º/2) / cos (120º/2)
3 2

 3  1.7
1 2
Chap 18. Molecular interactions
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• A better approach to the calculation of dipole
moments is to take into account the locations and
magnitudes of the partial charges on all the atoms.
These partial charges are included in the output of
many molecular structure software packages,
• To calculate the x-component, for instance, we need
to know the partial charge on each atom and the atom
x-coordinate relative to a point in the molecule and
form the sum
µx= ∑J qJ xJ
• Here qJ is the partial charge of atom J, xJ is the xcoordinate of atom J, and the sum is over all the
atoms in the molecule.
Chap 18. Molecular interactions
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• When the two dipole moments have the same
magnitude (as in the dichlorobenzenes), this equation
simplifies to
• µ = (µx2 + µy2 + µz2 ) 1/2
Chap 18. Molecular interactions
Example 18.1
Calculating a molecular dipole moment
• Estimate the electric dipole moment of the amide group
shown in (5) by using the partial charges (as multiples
of e ) in Table 18.2 and the locations of the atoms
shown.
Method
We use eqn 18.3a to calculate each of the components
of the dipole moment and then eqn 18.3b to assemble
the three components into the magnitude of the dipole
moment. Note that the partial charges are multiples of
the fundamental charge, e =1.609x10-19 C.
Chap 18. Molecular interactions
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29
Chap 18. Molecular interactions
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Chap 18. Molecular interactions
31
-
+X
-
+Y
Chap 18. Molecular interactions
-
Answer
32
• The expression for µx is :
µx = (-0.36 e) x (132 pm) + (0.45 e) x (0 pm) + (0.18 e) x
(182 pm) + (-0.38 e) x (-62.0 pm)
= 8.8 e pm
= 8.8 e pm x (1.609x10-19 C/e)x(10-12 m/pm)
= 1.4x10-30 C m
• Corresponding to µx = 0.42 D. The expression for µy is
µy = (-0.36 e) x (0 pm) + (0.45 e) x (0 pm)
+ (0.18 e) x (-86.6 pm) + (-0.38 e) x (107 pm)
= -56 e pm = -9.1x10-30 C m
Chap 18. Molecular interactions
33
• We can find the orientation of the dipole moment by
arranging an arrow of length 2.7 units of length to
have x, y, and z components of 0.42, -2.7, and 0
units; the orientation is superimposed on (6).
µ = (µx2 + µy2 + µz2 ) 1/2
µ 2= (0.42)2+(-2.7)2+02 = 7.466
µ = 2.7 D
Chap 18. Molecular interactions
Self-test 18.2
• Calculate the electric dipole moment of formaldehyde,
using the information in (7).
• The expression for µx is :
µx = (0.02 e) x (0 pm) + (-0.38 e) x (0 pm) + (0.18 e) x
(+94 pm) + (0.18 e) x (-94 pm) = 0.0 e pm
= 0.0 e pm x (1.609x10-19 C/e)x(10-12 m/pm)
=0Cm
µy = (0.02 e) x (0 pm) + (-0.38 e) x (118 pm) + (0.18 e)
x (-61.0 pm) + (0.18 e) x (-61.0 pm) = -44.8 e pm
= -66.80 e pm x (1.609x10-19 C/e)x(10-12 m/pm)
= 1.075x10-30 C m
• [-3.2 D]
Chap 18. Molecular interactions
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35
Chap 18. Molecular interactions
(b) Polarization
• Polarization (P ) of a bulk sample is the electric dipole
moment density, the mean electric dipole moment of
the molecules, the mean electric dipole moment of the
molecules, <µ >, multiplied by the number density,
N=N/V
• P = <µ>N
(18.4)
• In the following pages we refer to the sample as a
dielectric, by which is meant a polarizable,
nonconducting medium.
Chap 18. Molecular interactions
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37
• In the absence of an applied field, the individual
molecule dipoles point in random direction, so the net
dipole moment per unit volume is zero. An isotropic
fluid sample has the polarization of zero since the
molecules adopt any random orientations, so <μ> =0.
• In the presence of a field, the dipoles become partially
aligned because of thermal motion. Some orientations
have lower energies than others. So the electric dipole
moment density is nonzero. At temperature T, with
the field E applied along z direction:
 2E
Z 
3kT
• We show in the Justification below that, at a
temperature T
Chap 18. Molecular interactions
Justification 18.1
• If the energy of the dipole in the field is
U(θ) = - μ•Ε = -μΕcos θ
• The number of molecule with potential energy U a dipole
has an orientation in the range θto θ+dθ is given by the
Boltzmann distribution,
N    N 0 e E cos kT
• When the exponent is small: N    N 0 1 E cos kT 
• Each molecule contributes a term μcos θ to the net dipole
moment in the direction of the electric field. The
polarization due to the orientation of dipole is estimated
by multiplying N(θ) by μcos θ and integrating
dP   cos  e E cos kT d


0
 cos  e E cos kT d
Chap 18. Molecular interactions
38
39
Justification
• Energy barrier of the dipole reorientation in the field is
E(θ) = -μΕcosθ
• The probability dP that a dipole has an orientation in the
range θto θ+dθ is given by the Boltzmann distribution,
dP  e -E   kT sin  d


0
e -E   kT sin  d
• The average value of the parallel dipole moments is

Z    cos  dP   cos  dP 
  e -E   kT sin  cos d
0


e -E   kT sin  d
E
-E   kT  E cos  kT  x cos   xy
x
, y  cos 
kT

1
  e x cos sin  cos  d   ye xy dy
Z  0 
 11
x cos  sin  d
xy dy
e
e


0
0
1
Chap 18. Molecular interactions
40
Justification
• Apply the integrals:
x  e x
e
xy dy 
e
1
x
1
• then
Z
1

ye xy
1
e x  e x e x  e x
dy 

2
x
x
x  e x
x  e x
e
e


1


2
x  e x
  ye xy dy   x
x
e
1


1
 1

  x
 
x

x

x
xy dy
e  e  x
x
e e
e

1
• Use the Langevin function L (x ):
 e x  e x 1  x
L x    x





 e  e x x  3
• we have
Z
with x  1
2
 e x  e x 1 
E
E
   x
   

x
x
3kT 3kT
 e e
Chap 18. Molecular interactions
41
• Langevin function L (x)
The Langevin function L (x) is the following mathematical function
It occurs in the expression for the mean electric dipole moment of a
system of polar molecules that are free to rotate in an electric field
of strength E at a temperature T,
< µ > = µ L (µ E /kT)
When L (x) << 1, L (x) = 1/3 x + ...
In this limit, the mean electric dipole moment < µ > = µ
This term contributes to the polarization of a medium.
2
E /3kT.
induced dipole moment µ * is the dipole induced by an applied
electric field of strength E
µ *=αE.
Chap 18. Molecular interactions
Dielectric
• Dielectric (介電材料)
A dielectric is a polarizable non-conducting medium.
• ferroelectric solid (鐵電物質)
A ferroelectric solid has a permanent polarization
because of a cooperative shift of some of its atoms in
a specific direction.
Chap 18. Molecular interactions
42
43
• When two charges q1 and q2 are separated by a
distance r in a vacuum, the potential energy of their
interaction is
q1q2
V 
4 or
• When the same two charges are immersed in a
medium (such as air or a liquid), the potential energy
is reduced to
q1q2
V 
4r
• where  is the permittivity of the medium. The
permittivity is normally expressed in terms of the
dimensionless relative permittivity, r (formerly and
still widely called the dielectric constant) of the
medium: r = / 0
Chap 18. Molecular interactions
Polarization of a dielectric
44
• The capacitance C of a capacitor is the ratio of the
charge Q on the conducting plates to the potential
difference ΔΦ between the plates:
Q
C 

• On the parallel-plate capacitors, if the space is
vacuum between the plates with area A and distance d,
the capacitance Co is given by:
 0A
Co 
d
Chap 18. Molecular interactions
Fig. 18.A1
Fig. 18.A1
Parallel-plate capacitor filled with a dielectric. The
plates are labeled A and A’; the dielectrics between the
plates.
Chap 18. Molecular interactions
45
46
• The permittivity of vacuum ε0
• ε0=1/μ0c
2
=8.854187817…x10-12 C2N-1m-2 (exactly).
• the permeability of vacuum μ0=4πx10-7 N A-2 (exact).
• If a dielectric is inserted between the plates, the
capacitance is increased considering the polarization of the
dielectric.
• When a dielectric is placed in an electric field, the electrons
in the molecules are pulled toward the positive plate and
the nuclei are pulled toward the negative plate. The center
of charge in molecules is shifted , the displacement
produces a layer of charges on the surfaces of opposite
charged plates. These surface charges reduce the electric
field strength within the dielectric due to the opposite field.
Chap 18. Molecular interactions
Fig. 18.A2
Fig. 18.A2
Representation of a
capacitor containing
dipolar molecules. The
molecules are shown in
their instantaneous
positions indicating the
partial orientation of the
dipoles.
Chap 18. Molecular interactions
47
48
• Dielectric in a capacitor is said to be polarized by the
applied E -field.
• Polarization P of dielectric is defined as the vector sum of
dipole moments µ i of individual
molecules per unit volume

P  i V
(P )
i
• If N molecule per unit volume with charges ±q separated by
δ, the magnitude of polarization vector (P ) is
P  Nq
• The surface charge density σpol is the charge per area,
which is equal to the dipole moment per unit volume :
P   pol  Q A
Chap 18. Molecular interactions
49
Electric field in the dielectric
• Capacitor with a dielectric between plates will change its
capacitance due to the change of E-field in dielectric .
• The surface charge density on the plates σfree are driven by
the external voltage on the wire, and will not change by
inserting the dielectric.
• Gauss’s law show that the electric field strength E in the
dielectric due to polarization is
 free   pol
E
ε0
• If there were no polarization, σpol =0, we have E = σfree /ε0
Since σpol =P, the electric field strength E is
 free  P
E
ε0
E   free ε 0
as  pol  0
• As P is produced by E, it can not use to calculate E.
Chap 18. Molecular interactions
Electric field in the dielectric
• As P ∝ E, we need to know how P depends on E.
P  el ε 0E
• The electric susceptibility (χel) of the dielectric between
plates is the proportional constant of polarization.

 el ε 0E
free
• The electric field strength E now is E 
ε0
E   free ε 0  ε 0 el 
• solve as
• So the electric field strength E in dielectric is reduced by
the surface charge of dielectric by a factor of 1 1 el 
• Now the potential of a capacitor filled with a dielectric is
  Ed   freed ε 0  ε 0 el 
ε
ε
A
r
0
• Capacitance is C   free A   ε 0  ε 0 el A d 
d
Chap 18. Molecular interactions
50
Electric field in the dielectric
• The relative permittivity (εr) of the dielectric is
ε r  1 el   C C 0
• which is so called the dielectric constant.
• Since the polarization is given by P  el ε 0E
• Now the dielectric constant of a medium as
P
εr  1 
ε 0E
• Measurements of the relative permittivity yield the
polarization information of the dielectric.
Chap 18. Molecular interactions
51
18.2 Polarizabilities
• A molecule under external electric field may redistribute its
charge and forms a induced dipole moment μ *.
• The dipole moment due to the distortion polarization is
polarizability α :
*  E
where α is the polarizability of a molecule.
• The molecular polarizability is a proportional tensor since the
dipole moment vector is not necessarily parallel to the electric
field strength vector. In general α is symmetric
• If the matrix is diagonalized, then αxy = αyz = αzx = 0
• In the principal axis system, the induced dipole moment is
parallel to the electric field vector, the mean electric polarizability
is (αxx + αyy + αzz)/3
• The substance is said to be isotropic if αxx = αyy = αzz and the
unit for polarizability is C2 m2 J-1 = C m / J C-1 m-1
Chap 18. Molecular interactions
52
Polarizability
• hyperpolarizability
If the response has nonlinear contribution, then the
additional term are referred to as hyperpolarizability β
of the molecule.
1
*  E  E 2  
2
• (a) Polarizability volume
Molecular polarizabilities are often expressed in terms
of the polarizabilities volume, α＇, where
α’= α / 4πεo.
• The dimensions of polarizability volume are those of
volume and its magnitude is similar to that of the
volume of the molecule.
Chap 18. Molecular interactions
53
18.2 Polarizability of a dielectric
54
• The electric polarizability are often in the form:

  ' E 
E
4 0
• α’ has a unit of volume as a03 ~ (C2 m2 J-1)/ (J-1 C2 m-1)
• The macroscopic polarizability of a bulk sample is the
number per unit volume (N/V) multiplied by the dipole
moment αE from the molecular property α.
N
P  E
V
Chap 18. Molecular interactions
55
Justification 18.2
• The change in energy when the field increases from 0 to E is
E
E
0
0
E     * dE   
1 2
EdE   E
2
• The contribution to the hamiltonian when a dipole is exposed
to an electric field in the z-direction is
H 1   Z E
• Use second order perturbation theory
E 2  
n
n
 *
H 1
*
2
n
E 00   E n0 
2

z ,0n
2
 E  0 
0 
E

E
n
0
n
• The polarizability of the molecule in the z-direction is
2
z ,0n
2e 2R 2

  2 0 
0 

E
E
E
n
n
o
Chap 18. Molecular interactions
(b) Polarization at high frequency
• There are three kinds of polarization under an
electric field:
Orientation polarization: From the permanent
dipoles.
Distortion polarization: From the distortion of
nucleus position.
Electronic polarization: From the distortion of
the electron distribution.
Chap 18. Molecular interactions
56
Polarization
• Orientation polarization :
The polarizability of a molecule varies with frequency of
the applied field. At low frequencies, there are orientation,
distortion, and electronic contributions. The orientation
polarization is the polarization that stems from the
alignment of molecular dipoles in the applied field. This
contribution is lost above about 1011 Hz.
• Distortion polarization :
The next contribution to disappear after polarization is the
distortion polarization, the polarization that stems from
the distortion of the locations of the nuclei. This
contribution disappears when the frequency exceeds the
vibrational frequency of the molecule.
Chap 18. Molecular interactions
57
58
• Electronic polarizability :
At optical frequencies, only the electronic polarization, the
polarization stemming from the shift in positions of the
electrons, remains. This contribution also vanishes at
frequencies in excess of electronic excitation frequencies.
Chap 18. Molecular interactions
59
Justification 18.3
• The polarizability of a molecule in the presence of an
electric field that is oscillating a frequency  in the zdirection is obtained using time-dependent
perturbation theory and is
2
2 n 0 z ,0n
 ω   2
 n n 0  2
• provided that  is not close to 
n0,
• As  → o, the eq. reduced to a static polarizability.
• As  → ∞, the polarizability becomes
 ω  
2
 2
n

n
2
0
z ,0n  0
Chap 18. Molecular interactions
as   
18.3 Relative permittivities
• Permittivity
The permittivity, ε, of a modified Coulomb potential in the
medium from q/4πεo r to q/4πεrεo r; where ε0 is the
vacuum permittivity, the latter is related to the vacuum
permittivity by
ε0 µ
0
c2 =1
The numerical value (which is based on the defined values
of c and µ 0) is ε0 = 8.854187816 x 10-12 J-1 C2 m-1.
relative permittivity, εr , of a medium (formerly dielectric
constant) is defined as
εr = ε / ε0
Chap 18. Molecular interactions
60
Reduction for electrolyte solutions
• The relative permittivity can have a very significant
effect on the strength of the interactions between ions
in solution. For instance, water has a relative
permittivity of 78 at 25°C, so the interionic Coulombic
interaction energy is reduced by nearly two orders of
magnitude from its vacuum value.
Chap 18. Molecular interactions
61
Debye equation
• The relative permittivity of a substance is large if its
molecules are polar or highly polarizable. The
quantitative relation between the relative permittivity
and the electric properties of the molecules is
obtained by considering the polarization of a medium,
and is expressed by the Debye equation:
 r  1  Pm

r  2
M
Chap 18. Molecular interactions
62
Debye equation
• The Debye equation gives the relation between the
relative permittivity and the electric properties of a
molecule
ε r  1 Pm
εr  2

M
• where M is the molar mass and ρ is the mass density
of the sample. Pm is the molar polarization of the
sample:
NA 
2 
  

Pm 
3εo 
3kT 
• The term μ 2/3kT from the thermal averaging of the
electric dipole moment in the presence of the applied
field. The corresponding expression without the
contribution from the permanent dipole moment is
called the Clausius-Mossotti equation.
Chap 18. Molecular interactions
63
64
• Clausius-Mossotti equation
• When there is no contribution to the polarization from
permanent electric dipole moments the ClausiusMossotti equation is used
ε r  1 N A

ε r  2 3Mε 0
• The Clausius-Mossotti equation is used when there is
no contribution from permanent electric dipole
moments to the polarization, either because the
molecules are nonpolar or because the frequency of
the applied field is so high that the molecules cannot
orientate quickly enough to follow the change in
direction of the field.
Chap 18. Molecular interactions
Example 18.1
65
• The relative permittivity of 1H35Cl(g)
• Give the polarizability and dipole moment of HCl(g) in table
18.A2, calculate the relative permittivity at 1 atm and 273 K.
2
εr 1 N 

 
ε r  2 3Vε 0 
3kT
2 
 N AP 
 
  1.456x 103

3kT 
 3RTε 0 
1 21.456x 103 
εr 

1
.
0041
1 1.456x 103 
Chap 18. Molecular interactions
Orientation polarization of a dielectric
• Polarization due to the orientation of dipole is given by
N  2E
P 
V 3kT
• The magnitude of the polarization of a dielectric in an
electric field is the sum of the distortion polarization and
the orientation polarization:
N
P 
V
2

 
3kT


E i

E i : average local electric field
• The local filed is not the same as the applied field since the
molecular dipole also produce a field.
E i  E ε r  2 / 3
• Under a low concentration,
2

N

 ε r  2 
• Now
P   

E
V 
3kT  3 
Chap 18. Molecular interactions
66
67
• Rearrange:
N
P 
V
 2  ε r  2 

E

 
3kT  3 

P
and ε r  1
ε 0E
• yield the Clausius-Mossotti equation
2 
εr 1 N 

 

ε r  2 3Vε 0 
3kT 
• Replace the number density with molar mass M, (N/V =
nNA/V=ρNA/M) the molar polarization (Pm) is
M
Pm 

 εr  1  N A 
2 

 
 

3kT 
 ε r  2  3ε 0 
• The unit of Pm is a molar volume
• The macroscopic Pm now connected to molecular properties
αand μ, measure of molar polarization Pm at varies
temperature gives α and μ.
Chap 18. Molecular interactions
68
M
Pm 

 εr  1  N A 
2 

 
 

3kT 
 ε r  2  3ε 0 
1/T
Dependence of the molar polarization (Pm) of
fluorobenzene on temperature.
Chap 18. Molecular interactions
69
N A
• The intercept is
, gives the molecular
3ε 0
polarizability as α=1.146x10-39 C2 m2 J-1
NA
• The slope is
, gives the dipole moment as
9ε 0k
2
µ=5.26x10-30 C m.
• The dipole moment of molecules are often expressed
in terms of Debye (D). 1 D=3.336x10-30 C m. Thus the
dipole moment of fluorobenzene is 1.57 D. (Table
18.2)
Chap 18. Molecular interactions
Example 18.2
• The relative permittivity of a substance is measured by
comparing the capacitance of a capacitor with and without
the sample present (C and C0 , respectively) and using
εr = C / C0 . The relative permittivity of camphor (8) was
measured at a series of temperatures with the results given
below. Determine the dipole moment and the polarizability
volume of the molecule.
Chap 18. Molecular interactions
70
Example 18.2
Method
Equation 18.14 implies that the polarizability and
permanent electric dipole moment of the molecules in
a sample can be determined by measuring εr at a
series of temperatures, calculating Pm, and plotting it
against 1/T. The slope of the graph is NAμ2/9εok and
its intercept at 1/T= 0 is NAα/3εok . We need to
calculate (εr -I)/(εr + 2) at each temperature, and then
multiply by M/ρ to form Pm.
Chap 18. Molecular interactions
71
72
Example 18.2
εr  1
Method According to eqn 15, we need to calculate
εr  2
at each T, and then multiply by M/ρ to form Pm,
θ/℃
ρ/(g cm-3)
εr
0
0.99
12.5
20
0.99
11.4
40
0.99
10.8
60
0.99
10.0
80
0.99
9.50
100
0.99
8.90
120
0.97
8.10
140
0.96
7.60
160
0.95
7.11
200
0.91
6.21
Chap 18. Molecular interactions
73
Example 18.2
• For Camphor, M =152.23 g mol-1,
(εr-1)/(εr+2)
Pm/(cm3 mol-1)
θ/℃
(103K)/T
0
3.66
12.5
20
3.41
11.4
0.776
119
40
3.19
10.8
0.766
118
60
3.00
10.0
0.750
115
80
2.83
9.50
0.739
114
100
2.68
8.90
0.725
111
120
2.54
8.10
0.703
110
140
2.42
7.60
0.688
109
160
2.31
7.11
0.670
107
200
2.11
6.21
0.634
106
εr
0.793
122
The intercept lies at 83.52 and the slope is 10.55
Chap 18. Molecular interactions
74
The intercept lies at 83.5 and
the slope is 10.6
Chap 18. Molecular interactions
75
N A
• The intercept lies at 83.5 =
, so α’=3.3x10-23 cm3
3ε 0
NA
• The slope is 10.6 =
, gives the dipole moment
9ε 0k
2
as µ = 4.39x10-30 C m, corresponding to 1.32 D.
• Because the Debye equation describes molecules that
are free to rotate, the data show that camphor, which
does not melt until 175°C, is rotating even in the
solid. It is an approximately spherical molecule.
Chap 18. Molecular interactions
Self-test 18.3
• Determination of the polarizability volume
• The relative permittivity of chlorobenzene is 5.71 at
20℃ and 5.62 at 25 ℃. Assuming a constant density
(1.11 g cm3), estimate its polarizability volume and
dipole moment.
• [M = 112.5 g cm-3; slope=8.142,α’= 1.35 x 10-23
cm3; intercept=34.1404; µ = 1.16 D]
Chap 18. Molecular interactions
76
Maxwell equations
• The Maxwell equations that describe the properties of
electromagnetic radiation relate the refractive index at
a (visible or ultraviolet) specified wavelength to the
relative permittivity at that frequency:
• nr = εr1/2
• Therefore, the molar polarization, Pm, and the
molecular polarizability, α, can be measured at
frequencies typical of visible light (about 1015 to 1016
Hz) by measuring the refractive index of the sample
and using the Clausius-Mossotti equation.
Chap 18. Molecular interactions
77
Interactions between molecules
• The van der Waals forces are the weak, attractive
intermolecular forces arising from permanent or induced
electric dipole moments. A van der Waals molecule is a
cluster of closed-shell atoms or molecules that are held
together by van der Waals forces that depends on the
distance between the molecules as l/r 6.
• In addition, there are interactions between ions and the
partial charges of polar molecules and repulsive interactions
that prevent the complete collapse of matter to nuclear
densities.
• The repulsive interactions arise from Coulombic repulsions
and, indirectly, from the Pauli principle and the exclusion of
electrons from regions of space where the orbitals of
neighbouring species overlap.
Chap 18. Molecular interactions
78
18.4 Interactions between dipoles
• When two charges q1 and q2 are separated by a distance r in a
vacuum, the potential energy of interaction is
q1 q 2
V r  
4ε 0r
• When a dipole μ1 = q1 ℓ and a point charge q2 is close to each
other, the potential energy of interaction is
1 q 2
V r  
4ε 0r 2
• Sum of the repulsion of like charges and the attraction of
opposite charges given the total potential energy of interaction.
For a point dipole, l  r.
Chap 18. Molecular interactions
79
80
The potential energy of
interaction between a dipole
and a point charge is the sum
of the repulsion of like
charges and the attraction of
opposite charges. For a point
dipole, l  r.
Chap 18. Molecular interactions
81
Justification 18.4
The interaction between a point charge and a point dipole
The sum of the potential energies of repulsion between
like charges and attraction between opposite charges in
the orientation shown in (9) is
Answer The sum of two contributions with x = ℓ/2r is


1  q1 q 2 q1 q 2  q1 q 2


V r  

4ε 0  r   r    4ε 0r


2
2

1 
 1



1 x 1 x 
Because ℓ « r for a point dipole, this expression can be
simplified by expanding the terms in x and retaining
only the leading term:
q1 q2
q1 q2 l ٛ
2xq1 q2
2
2

V r  

1  x  x  ...  1  x  x  ...  
4ε 0r
2ε 0r 2
4ε 0r


Chap 18. Molecular interactions
82
Chap 18. Molecular interactions
83
The potential energy of interaction between two
dipoles is the sum of the repulsions of like charges
and the attractions of opposite charges. This
illustration shows a collinear arrangement of dipoles.
Chap 18. Molecular interactions
84
Example 18.3
• Calculating the interaction energy of two dipoles
• Calculate the potential energy of interaction of two
dipoles in the arrangement shown in Fig 18.9 when
their separation is r and θ =0
Answer The sum of four contributions with x = ℓ/r is
1  q1 q 2 q1 q 2 q1 q 2 q1 q 2 
q1 q 2



V r  


4ε 0  r  
r
r
r  
4ε 0r
1 
 1
2


1 x 
1 x
The expansion of two terms in x and only the first
surviving term retains, which is 2x 2. Now with μ1 =
q1ℓ and μ2 = q2ℓ, the potential energy
2
2x q1 q 2
1 2

V r   
3
4ε 0r
2ε 0r
Chap 18. Molecular interactions
85
A parallel arrangement
of electric dipoles as
used for Self-test 18.4
Chap 18. Molecular interactions
86
Self-test 18.4
• Derive an expression for the potential energy when the
interaction of two dipoles are in the arrangement shown in
(11) when their separation is r and θ =90°
Answer The sum of four contributions with x = ℓ/r is
q1 q 2
q1 q 2
q1 q 2
1  q1 q 2

V r  



r
4ε 0  r
r 2  2
r 2  2
d 2 = (r 2+ℓ2) = r 2(1+x2 ) ≈ r
2
 2q1 q 2 
1
 
1 
2
ε
r

4

1
x
0 

(1+ x2/2 - x4/8 +… )2
Now with μ1 = q1ℓ and μ2 = q2ℓ, retains term x 2. theV (r ) is
2
4 1 

1 2  r 
1 2   x
x  

1  
1  1 
V r  

2 
2
2ε 0r   d  2ε 0r   
2
8  


2
4
4


1 2 
x
x
x   1 2
  
1  1 



2 
2ε 0r   
2
8
4   4ε 0r 3
Chap 18. Molecular interactions



87
For this arrangement,
the expression should
be multiplied by cosθ
when the dipole lies at
an angle θ to the axis of
the dipole.
Chap 18. Molecular interactions
88
Chap 18. Molecular interactions
Multipole
89
• Monopole : is a point charge
• Dipole :
A point dipole is an electric dipole in the limit R → 0
• Multipole
An electric multipole is an array of point charges, of
which an electric dipole, two equal and opposite point
charges, is a special case.
• n-pole
Specifically, an electric n-pole is an array of n point
charges with an n-pole moment but no lower moment.
A single point charges with zero overall charge and no
dipole moment is called a quadrupole.
Chap 18. Molecular interactions
90
Chap 18. Molecular interactions
91
• An octupole consists of an array of point charge that
sum to zero and which has neither a dipole moment
nor a quadrupole moment, (as for CH4 molecule).
• The interaction energy falls off more rapidly the
higher the order of the multipole.
• The n-pole interacting with an m-pole, the potential
energy varies with distance as
V r  
1
4ε 0  r
n m 1
Chap 18. Molecular interactions
92
Typical charge arrays corresponding to electric multipoles.
The field arising from an arbitrary finite charge distribution
can be expressed as the superposition of the fields arising
from a superposition of multipoles.
Chap 18. Molecular interactions
93
(b) Electric field
• The potential energy of a charge q1 in the presence of
another charge q2 is V (r )= q1φ where φ is the
Coulomb potential.
q1 q2
• V (r ) =q1 
, and the electric field is E  
4ε 0 r
q2
• The electric field around a point charge q2 is E 
2
4ε 0 r
• For a point charge q1 close to a dipole μ2 = q2 ℓ, the
q1 2
potential energyV r  
2
4ε 0r
• So the electric field around a dipole μ2 is E 
2
3
4ε 0 r
• The electric field of a multipole decreases more
rapidly with distance than a monopole (a point charge).
Chap 18. Molecular interactions
94
Chap 18. Molecular interactions
Further Information 18.1 dipole-dipole interaction
• Potential energy of interaction between two dipoles
• The potential energy of μ2 in the electric field E1 that
generated by μ 1 is given by the dot (scalar) product:
1  2
V r  
 E1  2
2
4ε 0r
• Consider a distribution of charges qi located at xi yi zi
from the origin. The Coulomb potential φ at r is
qi
qi
 r   


1
2
/
2
2
2
i 4ε 0 x  x i   y  y i   z  z i 
i 4ε 0 r  ri 
• Assume all charges are close to the origin (or ri <<r ),
use a Taylor expansion:

qi  1   r  ri 
qi  1 xx i

   
 r   
 
  3  

ri r 0
r

i 4ε 0 r
i 4ε 0 r




i


Chap 18. Molecular interactions
95
96
(c) Dipole-dipole interaction
• If the charges is electrically neutral, ∑qi =0, the first
term disappear. And ∑qixi =μx, ∑qiyi =μy, ∑qizi = μz
 r  
1
4ε 0 r
3
x x  y y  z z  
1
4ε 0 r
3
μ 1 r
• The electric field strength is E1   r 
1
 1r
1
μ 1 r 1
E1  
 3 

 3
3
r
4ε 0
4ε 0 r
4ε 0 r
• If follows that

1 2
μ 1  r  μ 2  r 
V r   E 1   2 
3
3
4ε 0 r
4ε 0 r 3
• with μ1·r = μ1r cosθ and μ2·r = μ2r cosθ
1 2 f  
2
V r  
f    1  3 cos 
3
4ε 0 r
Chap 18. Molecular interactions
Interaction between two dipoles
• The potential energy of interaction between two polar
molecules in a fixed, parallel, orientation in a solid:
1 2 f  
V r  
3
4ε 0 r
f    1  3 cos 
2
Chap 18. Molecular interactions
97
98
Comment 18.9
Potential energy of two dipoles based on their orientation:
 For two parallel dipoles in fixed orientation in a solid
1 2 f  
2


V r  
f


1

3
cos

3
4ε 0 r
 In the fluid of freely rotating molecules, the
interaction between dipoles averages to zero.
 The average value of a function f (x) over range x=a
b
1
to x=b is f x  
f x  dx

b a a
 In polar coordinates, the volume element is
proportional to sinθ dθ , therefore the spherical
average value of <1-3cos2θ > over 0 to π is zero.
1 
2
2
1  3 cos  
1

3
cos
 sin  d  0

 0 0


Chap 18. Molecular interactions
99
• The interaction energy of two freely rotating dipoles is
zero. However, because their mutual potential energy
depends on their relative orientation, in fact the
molecules do not rotate completely freely, even in a
gas.
• In fact, the lower energy orientations are marginally
favored, so there is a nonzero average interaction
between two rotating polar molecules with moments
μ1 and μ2 separated by a vector r.
• Keesom interaction: the nonzero average potential
energy of two rotating polar molecules :
2 2
C
2 1 2
V  6 C 
2
r
34ε 0  kT
Chap 18. Molecular interactions
Justification 18.5 The Keesom interaction
100
• The average interaction potential of two polar molecules
rotating at a fixed separation r is given by
1 2 f  
V r  
3
4ε 0 r
where <f (θ)> includes a weighting factor in the averaging
that is the probability of a particular orientation. From
Boltzmann distribution, P ∝ e-E/kT here E as potential V
1 2 f
V / RT
P e
V r  
4ε 0 r 3
when V is small compare to thermal motion :V << kT ;
in the exponential function P ∝ (1-V/kT + …); the
weighed average of <f (θ )> is
1 2
2
f    f   0 
f    ...
3
0
4 0kT r
• where < ... >0 denotes an unweighted spherical average.
Chap 18. Molecular interactions
101
• The spherical average of f is zero, so the first term
< f (θ )>0 vanishes.
• However, the average value of f
positive at all orientations
V r   
2
is nonzero because f is
1222 f 2   0
4ε 0  kT r
2
6
• The average value <f 2>0 turns out to be 2/3 when the
calculation is carried through in detail. The final result is
that quoted as
C
V  6
r
2 
C 
34ε 0  kT
2
1
Chap 18. Molecular interactions
2
2
2
Keesom interaction
102
• The important features of Keesom interaction are
 negative sign, (the average interaction is attractive)
 dependence of the average interaction energy on the
inverse sixth power of the separation, 1/r 6, (which
identifies it as a van der Waals interaction),
 inverse dependence on the temperature.
• The last feature reflects the way that the greater thermal
motion overcomes the mutual orientating effects of the
dipoles at higher temperatures.
• The 1/r 6 arises from the 1/r 3 of the interaction
potential energy that is weighted by the energy in the
Boltzmann term, which is also proportional to 1/r 3.
Chap 18. Molecular interactions
103
• At 25℃ the average interaction energy for pairs of
molecules with μ =1 D is about -0.07 kJ mol-1 when
the separation is 0.5 nm. This energy should be
compared with the average molar kinetic energy of
3RT/2 = 3.7 kJ mol-1 at the same temperature.
• The interaction energy is also much smaller than the
energies involved in the making and breaking of
chemical bonds.
Chap 18. Molecular interactions
(d) Dipole-induced dipole interaction
• A polar molecule with dipole moment μ 1 can induce a
dipole μ 2* in a neighboring polarizable molecule. The
partial charges of the former give rise to an electric
field that polarizes the second molecule, The induced
dipole interacts with the permanent dipole of the first
molecule, and the two are attracted together.
• With first dipole moment μ1 and the polarizability
volume α2’ for the second molecule:
C
V r    6
r

C 
4ε 0
2
1
'
2
• which is temperature independent because thermal
motion has no effect on the averaging process.
Chap 18. Molecular interactions
104
105
Chap 18. Molecular interactions
106
• The potential energy depends on 1/r 6, like the
dipole-dipole interaction,
• this distance dependence stems from the 1/r 3
dependence of the field (and hence the magnitude of
the induced dipole) and the 1/r 3 dependence of the
potential energy of interaction between the permanent
and induced dipoles.
• For a molecule with μ = I D (such as HCl) near a
molecule of polarizability volume α’ = 10 x 10-30 m3
(such as benzene), the average interaction energy is
about -0.8 kJ mol-1 when the separation is 0.3 nm.
Chap 18. Molecular interactions
(e) Induced dipole-induced dipole interaction:
107
An induced dipole-induced dipole interaction is also
known as a London interaction and a dispersion
interaction. It arises when a fluctuation in the electron
density of one (possibly non-polar) molecule results
in the formation of an instantaneous dipole, that
dipole induces a dipole in a neighboring molecule,
and the two instantaneous dipoles attract each other
and the potential energy of the pair is lowered. The
direction and magnitude of the dipoles are correlated,
so the interaction does not average to zero. (figure
18.7)
Chap 18. Molecular interactions
108
Chap 18. Molecular interactions
109
• Polar molecules also interact by a dispersion
interaction: such molecules also possess
instantaneous dipoles, the only difference being that
the time average of each fluctuating dipole does not
vanish, but corresponds to the permanent dipole.
• Therefore the molecules interact both through their
permanent dipoles and through the correlated,
instantaneous fluctuations in these dipoles.
• The strength of the dispersion interaction depends on
the polarizability of the first molecule because the
instantaneous dipole moment μ 1* , depends on the
looseness of the control that the nuclear charge
exercises over the outer electrons.
Chap 18. Molecular interactions
London formula
• The strength of the interaction also depends on the
polarizability of the second molecule, for that
polarizability determines how readily a dipole can be
induced by another molecule. a reasonable
approximation to the interaction energy is given by
the London formula:
<V> = -C/r 6
where C = 2 a1’a2’I1 I2 /3 (I1 + I2)
• and I1 and I2 are the ionization energies of the
molecules.
• This interaction energy is also proportional to 1/r6 of
the separation of the molecules, which identifies it as
a third contribution to the van der Waals interaction.
Chap 18. Molecular interactions
110
Illustration 18.1
111
• Calculating the strength of the dispersion interaction :
• For two CH4 molecules, we can substitute α' =2.6 x10-30
m3 and I ≈700 kJ mol-1 to obtain V = -2 kJ mol-1 for r =
0.3 nm. A very rough check on this figure is the enthalpy
of vaporization of methane, which is 8.2 kJ mol-1.
However, this comparison is insecure, partly because the
enthalpy of vaporization is a many-body quantity and
partly because the long-distance assumption breaks
down.
Chap 18. Molecular interactions
hydrogen bond
112
• A hydrogen bond is a link between two molecules (or
internally within a molecule) mediated by a proton lying
between two highly electronegative non-metallic
elements, specifically N, O, and F. Hydrogen bonds may
also form to anions.
• A hydrogen bond is the strongest of the intermolecular
interactions, corresponding to about 20 kJ mol-1. Its
presence lowers the vapor pressure of substances and
provides rigid structures in solids. The viscosities of
liquids are increased when hydrogen bonding is
possible, and the bonding contributes to the hydration
of ions in solution. The numerous anomalous properties
of water can almost all be traced to the presence of
hydrogen bonds between the molecules.
Chap 18. Molecular interactions
hydrogen bond
113
• A variety of explanations can be given for the formation
of the bond, but all depend on the small size of the
proton and its ability to lie between two other atoms.
• In the molecular orbital explanation, the atoms A, H, and
B each supply one atomic orbital from which molecular
orbitals covering the AHB fragment can be built, and from
these three atomic orbitals three molecular orbitals can
be formed. One orbital is bonding (b), one approximately
non-bonding (n), and the third anti-bonding (a).
• There are four electrons to accommodate (two from the
A-H bond, two from the B-lone pair), the configuration of
the fragment is b2n2, because the antibonding orbital is
unoccupied, the fragment may be lower in energy than
the separated components.
Chap 18. Molecular interactions
114
ψ = c1 ψ A + c2 ψ H + c3 ψ
Chap 18. Molecular interactions
B
(g) The hydrophobic interaction
115
• Nonpolar molecules do dissolve slightly in polar
solvents, but strong interactions between solute and
solvent are not possible and as a result it is found that
each individual solute molecule is surrounded by a
solvent cage.
• Consider the thermodynamics of transfer of a
nonpolar hydrocarbon solute from a nonpolar solvent
to water, a polar solvent.
• Experiments indicate that the process is endergonic
(∆transferG > 0), as expected on the basis of the
increase in polarity of the solvent, but exothermic
(∆transferH < 0). Therefore, it is a large decrease in the
entropy of the system (∆transferS < 0) that accounts for
the positive Gibbs energy of transfer.
Chap 18. Molecular interactions
116
Chap 18. Molecular interactions
117
• For example, the process
CH4(in CCl4 ) → CH4 (aq)
has ∆transferG =+12 kJ mol-1, ∆transferH = -10 kJ mol-1,
and ∆transferS = -75 J K-1mol-1 at 298 K.
• Substances characterized by a positive Gibbs energy
of transfer from a nonpolar to a polar solvent are
called hydrophobic.
Chap 18. Molecular interactions
hydrophobicity constant, π
118
• It is possible to quantify the hydrophobicity of a small
molecular group R by defining the hydrophobicity
constant, π, as
π = log (S/S₀)
where S is the ratio of the molar solubility of the
compound R-A in octanol, a nonpolar solvent, to that in
water, and S₀ is the ratio of the molar solubility of the
compound H-A in octanol to that in water.
• Positive values of π indicate hydrophobicity and
negative values of π indicate hydrophilicity, the
thermodynamic preference for water as a solvent. It is
observed experimentally that the π values of most
groups do not depend on the nature of A.
Chap 18. Molecular interactions
119
• For example:
• R = CH3, CH2CH3, (CH2)2CH3, (CH2)3CH3, (CH2)4CH3
• π = 0.5,
1.0,
1.5,
2.0,
2.5
and we conclude that acyclic saturated hydrocarbons
become more hydrophobic as the carbon chain length
increases.
• This trend can be rationalized by ∆transferH becoming
more positive and ∆transferS more negative as the
number of carbon atoms in the chain increases.
Chap 18. Molecular interactions
At the molecular level
120
• Formation of a solvent cage around a hydrophobic
molecule involves the formation of new hydrogen bonds
among solvent molecules. This is an exothermic process
and accounts for the negative values of ∆transferH .
• On the other hand, the increase in order associated with
formation of a very large number of small solvent cages
decreases the entropy of the system and accounts for
the negative values of ∆transferS. However, when many
solute molecules cluster together, fewer (albeit larger)
cages are required and more solvent molecules are free
to move. The net effect of formation of large clusters of
hydrophobic molecules is then a decrease in the
organization of the solvent and therefore a net increase
in entropy of the system.
Chap 18. Molecular interactions
121
• This increase in entropy of the solvent is large enough
to render spontaneous the association of hydrophobic
molecules in a polar solvent.
• The increase in entropy that results from fewer
structural demands on the solvent placed by the
clustering of nonpolar molecules is the origin of the
hydrophobic interaction, which tends to stabilize
groupings of hydrophobic groups in micelles and
biopolymers (Chapter 19). The hydrophobic
interaction is an example of an ordering process that
is stabilized by a tendency toward greater disorder of
the solvent.
Chap 18. Molecular interactions
(h) The total attractive interaction
122
• We shall consider molecules that are unable to
participate in hydrogen bond formation. The total
attractive interaction energy between rotating
molecules is then the sum of the three van der Waals
contributions discussed above. (Only the dispersion
interaction contributes if both molecules are nonpolar).
• In a fluid phase, all three contributions to the potential
energy vary as the inverse sixth power of the
separation of the molecules, so we may write
V = -C6 /r
6
• where C6 is a coefficient that depends on the identity
of the molecules.
Chap 18. Molecular interactions
123
• Although this equation represents the attractive interactions
between molecules, it has only limited validity.
• First, we have taken into account only dipolar interactions of
various kinds, for they have the longest range and are
dominant if the average separation of the molecules is large.
However, in a complete treatment we should also consider
quadrupolar and higher-order multipole interactions,
particularly if the molecules do not have permanent electric
dipole moments.
• Secondly, the expressions have been derived by assuming that
the molecules can rotate reasonably freely. That is not the case
in most solids. and in rigid media the dipole-dipole interaction
is proportional to 1/r 3 because the Boltzmann averaging
procedure is irrelevant when the molecules are trapped into a
fixed orientation.
Chap 18. Molecular interactions
Axilrod-Teller formula
• A different kind of limitation is that the eqn relates to
the interactions of pairs of molecules. There is no
reason to suppose that the energy of interaction of
three (or more) molecules is the sum of the pairwise
interaction energies alone.
• The total dispersion energy of three closed-shell
atoms, for instance, is given approximately by the
Axilrod-Teller formula:
C6
C6 C6
C'
V  6 6 6
3
rAB rBC rCA rAB rBC rCA 
• where C’ = a (3 cos θA cos θB cos θC +1)
Chap 18. Molecular interactions
124
The Axilrod-Teller formula
Chap 18. Molecular interactions
125
Ailrod-Teller formula
126
• Ailrod-teller formula : This formula allows the calculation
of the interaction of three or more molecules
• V = C6/rAB6 - C6/rBC6 – C6/rCA6 + C ’/(rAB rBC rBC)3 where
C’ = a (3 cos θA cos θB cos θC +1)
• The parameter a is approximately equal to (¾) α'C6; the
angles θ are the internal angles of the triangle formed by
the three atoms (14). The term in C' (which represents
the non-additivity of the pairwise interactions) is
negative for a linear arrangement of atoms (so that
arrangement is stabilized) and positive for an equilateral
triangular cluster.
• It is found that the three-body term contributes about 10
per cent of the total interaction energy in liquid argon.
Chap 18. Molecular interactions
18.5 Repulsive and total interactions
127
• When molecules are squeezed together, the nuclear
and electronic repulsions and the rising electronic
kinetic energy begin to dominate the attractive forces.
• The repulsions increase steeply with decreasing
separation in a way that can be deduced only by very
extensive, complicated molecular structure
calculations.
• However, in many cases, progress can be made
by using a greatly simplified representation of the
potential energy, where the details are ignored and
the general features expressed by a few adjustable
parameters.
Chap 18. Molecular interactions
128
18.9a In a two-dimensional
molecular dynamics
simulation that uses
periodic boundary
conditions when one
particle leaves the cell its
mirror image enters
through the opposite face.
Chap 18. Molecular interactions
18.5 Repulsive and total interactions
• Hard-sphere potential
• One such approximation is the hard sphere potential,
in which it is assumed that the potential energy rises
abruptly to infinity as soon as the particles come
within a separation d:
• A hard-sphere potential is a model intermolecular
potential in which there is no interaction between two
particles until they are at a specified distance, when
the potential energy climb abruptly to infinity
• V = ∞ if r ≤ d
• V = 0 if r > d
Chap 18. Molecular interactions
129
22.4 Repulsive and total interactions
130
• Mie potential
• This is an approximation used for the intermolecular potential
n
m
energy
where n>m
C
C
   
V r       
r  r 
• Lennard-Jones potential
• The Lennard-Jones potential is a model intermolecular potential
energy that expresses the potential energy of two molecules as a
function of their separation r in terms of two parameters. The
n
6
Lennard-Jones (n,6)-potential is
C  C 
V r   
  
r  r 
• The first term (when n >6) expresses the repulsive contribution
to the potential energy, which is operative at short distances
(essentially contact) and the second term expressed the longer
range attractive interactions. Several attractive intermolecular
interactions are inversely proportional to r 6.
Chap 18. Molecular interactions
131
Chap 18. Molecular interactions
The Lennard-Jones potential
the relation of the
parameters to the features
of the curve, and the two
contributions. Note that
21/6 = 1.122 . . . .
  ro 12  ro  6 
V r   4       
 r 

r




ε : well depth
ro: closest distance when
V (r )=0
Chap 18. Molecular interactions
132
Lennard-Jones (12,6)-potential
133
• For mathematical reasons it is sometimes convenient
to use the Lennard-Jones (12,6)-potential, with n =12.
which is normally parametrized as follows:
  ro 12  ro 6 
V r   4       
r  
 r 
• The parameter ε is the depth of the potential well and
ro is the separation at which the curve cuts through
V (r )=0. The well minimum occurs at
r = 21/6 ro
• The exp-6 potential: V = 4ε{e–r/r0 – (r0/r)6}.
A potential with an exponential repulsive term and a
1/r 6 attractive term.
Chap 18. Molecular interactions
134
Chap 18. Molecular interactions
135
Chap 18. Molecular interactions
Atomic force microscopy
136
• The potential energy of interaction of molecules was
the primary interest in the molecular-size probes and
the surface..
• Atomic force microscopy (AFM): a technique in which
the force between a molecular sized probe and a
surface is monitored.
• As force, F, is the negative slope of potential,
13
7

dV r  24   ro 
 ro  

F 
2    
dr
ro   r   r  
Chap 18. Molecular interactions
The greatest attractive force
137
• For a Lennard Jones potential, the net attractive force
is greatest at a distance of r = 1.244σand with a force
F = -2.397ε/r o:
  ro 12  ro 6 
V r   4       
r  
 r 
2
 V r   4
12 13
6 7

 12ro r  6ro r
2
r
r
12 14
6 8
 4 156ro r  42ro r

 4r

14

156r
156r
 4r
29 6
6
r  ro
7
14
12
o
 42ro r 6
6
 42ro r
29
r  6 ro
7
12
o
Chap 18. Molecular interactions
6 6


 0
Molecular beam
138
• Molecular beam
A molecular beam is a narrow stream of molecules with
a narrow spread of velocities and in some cases, in
specific internal states or orientations. Molecular beam
studies of non-reactive collisions are used to explore
the details of intermolecular interations with a view to
determining the shape of the intermolecular potential.
Molecular beam studies of reactive collisions are used to
study the details of collisions that lead to the exchange
of atoms and hence to explore the shape and features of
the molecular potential energy surface. Surfaces,
particularly the ability of particular features to act as
sites of catalytic activity, are also studied using
molecular beams. (Figure)
Chap 18. Molecular interactions
139
Chap 18. Molecular interactions
140
Fig.18.13 A 3D QSAR analysis of the binding of steroids,
molecules with the carbon skeleton shown, to human
corticosteroid-binding globulin (CBG). The ellipses
indicate areas in the protein's binding site with positive
or negative electrostatic potentials and with little or
much steric crowding. It follows from the calculations
that addition of large substituents near the left-hand
side of the molecule (as it is drawn on the page) leads to
poor affinity of the drug to the binding site. Also,
substituents that lead to the accumulation of negative
electrostatic potential at either end of the drug are likely
to show enhanced affinity for the binding site.
(Adapted from P. Krogsgaard-Larsen, T. Liljefors, U. Madsen (ed.),
Textbook of drug design and discovery, Taylor & Francis, London
(2002).)
Chap 18. Molecular interactions
141
Chap 18. Molecular interactions
Effects of Molecular Interactions on Collisions
142
• The extent of scattering may depend on the relative
speed of approach as well as the impact parameter b.
Two paths leading to the same destination will
interfere quantum mechanically; in this case they
give rise to quantum oscillations in the forward
direction. Interference between the numerous paths
at that angle modifies the scattering intensity
markedly.
Chap 18. Molecular interactions
Effects of Molecular Interactions on Collisions
143
Potential Energy
The basic arrangement of a
molecular beam apparatus. The
atoms or molecules emerge
from a heated source, and pass
through the velocity selector, a
train of rotating disks. The
scattering occurs from the
target gas (which might take
the form of another beam), and
the flux of particles entering
the detector set at some angle
is recorded.
(a) The definition of the solid
angle, d, for scattering.
(b) The definition of the impact
parameter, b, as the
perpendicular separation of the
initial paths of the particles.
Chap 18. Molecular interactions
144
Chap 18. Molecular interactions
145
Chap 18. Molecular interactions
The impact parameter
• impact parameter
• The impact parameter, b, is the initial perpendicular
distance between a target molecule and the line of
approach of a projectile molecule. (Figure)
Chap 18. Molecular interactions
146
147
Chap 18. Molecular interactions
Effects of Molecular Interactions on Collisions
Potential Energy
Three typical cases for the collisions of two hard
spheres:
(a) b = 0, giving backward scattering;
(b) b  RA + RB, giving forward scattering;
(c)
0  b  RA + RB, leading to scattering into one
direction on a ring of possibilities.
(The target molecule is taken to be so heavy that it
remains virtually stationary.)
Chap 18. Molecular interactions
148
Effects of Molecular Interactions on Collisions
• Figure Collision of two
spherically symmetrical
molecules with an
impact parameter b.
The diagram is drawn
so that the center of
gravity of the system
does not shift during
the collision. The angle
of deflection is χ.
Chap 18. Molecular interactions
149
150
18.17 The extent of scattering
may depend on the relative
speed of approach as well as
the impact parameter. The dark
central zone represents the
repulsive core; the fuzzy outer
zone represents the long-range
attractive potential.
Chap 18. Molecular interactions
151
Chap 18. Molecular interactions
152
Chap 18. Molecular interactions
153
18.18 Two paths leading to the
same destination will interfere
quantum mechanically; in this
case they give rise to quantum
oscillations in the forward
direction.
Chap 18. Molecular interactions
154
• Quantum oscillation
• A particle with a certain impact parameter might
approach the attractive region of the potential in such
a way that the particle is deflected towards the
repulsive core, which then repels it out through the
attractive region to continue its flight in the forward
direction. Some molecules, however, also travel in the
forward direction because they have impact
parameters so large that they are un-deflected. The
wave functions of the particles that take the two types
of path interfere, and the intensity in the forward
direction is modified. This effect is called quantum
oscillation. (Figure)
Chap 18. Molecular interactions
Effects of Molecular Interactions on Collisions
155
Fig. 18.19 The dark central
zone represents the
repulsive core; the fuzzy
outer zone represents the
long-range attractive
potential. The
interference of paths
leading to rainbow
scattering. The rainbow
angle, r, is the maximum
scattering angle reached
as b is decreased.
Chap 18. Molecular interactions
156
• Rainbow scattering
• As the impact parameter b for a molecular collision is
decreased the angle of scattering increases, passes
through a maximum, and then decreases again. There
is strong constructive interference between
neighboring paths at the turning point, and hence a
pronounced peak in scattered intensity. The
phenomenon is termed rainbow scattering, and the
angle at which it occurs is the rainbow angle, θr,
(Figure)
Chap 18. Molecular interactions
157
Chap 18. Molecular interactions
Effects of Molecular Interactions on Collisions
• Collision trajectories
for a pair of molecules
interacting by a
Lennard-Jones 6-12
potential.
Chap 18. Molecular interactions
158
Surface Tension
159
• 表面張力（surface tension）在兩相(特別是氣-液)界面上，
處處存在著一種張力，它垂直與表面的邊界，指向液體方向並
與表面相切。
• 把作用於單位邊界線上的這種力稱為表面張力，用γ表示，單位
是 N/m.
• 氣-液界面的表面張力γ是製造單位界面(dAS) ，與所需功 dw
的比例常數 dw = γ dAS.
• 廣義的表面自由能定義：保持對應量的特徵自然變數不變，每
增加單位表面積時，對應量熱力學函數的增加值。
• 狹義的表面自由能定義： 保持溫度、壓力和組成不變，每增加
單位表面積時，Gibbs自由能(以功的形式)的增加值，稱為表面
Gibbs自由能，或簡稱表面自由能或表面能，單位為 J/m2 。
Chap 18. Molecular interactions
160
F  2γl; AS  2h l
F dh 2γl
dw


γ
dAS 2l dh 2l
Chap 18. Molecular interactions
Table 6.2 Surface Tensions of Some Liquids
Substance
t/℃
γ/mN m-1
Inorganic
Aluminum
30
840
Gold
40
1102
Lead
60
453
Mercury
100
472
Lead chloride
140
138
Sodium chloride
180
106
Oxygen
-183
13.1
Nitrogen
-183
6.2
Chap 18. Molecular interactions
161
Table 6.2 Surface Tensions of Some Liquids
Substance
t/℃
γ/mN m-1
Organic
Acetone
20
23.7
Benzene
20
28.88
Ethanol
20
22.8
Diethyl ether
20
16.96
Glycerol
20
63.4
Water
0
75.7
15
73.5
25
72.0
50
67.9
100
58.8
Chap 18. Molecular interactions
162
163
Chap 18. Molecular interactions
164
Chap 18. Molecular interactions
Curved surfaces
165
• 凸面(Convex)的曲率半徑取正值，凹面(Concave)的曲率半徑取負值。
• A curved surface of a liquid, or a curved interface between phases,
exerts a pressure so that the pressure is higher in the phase on the
concave side of the interface. (在液體屈面或是兩相的界面兩邊,施加於凹
面的內部壓力較外部壓力大)
• The pressure inside of a rubber balloon is higher than the
atmospheric pressure. (橡皮氣球的內部壓力比大氣壓力較高)
• Bubble is a region in which vapor is trapped by a thin film of liquid.
(液膜泡是平衡蒸氣包在液體薄膜中)
• Cavity is a vapor-filled hole in a liquid. (氣泡是在液體中的洞,填充了平
衡蒸氣壓)
• Droplet is a small volume of liquid at equilibrium with and
surrounded by its vapor. (液滴是小體積的液體被周圍的平衡蒸氣壓包圍)
Chap 18. Molecular interactions
166
Figure A
• Spherical droplet of
a pure liquid in
contact with its
vapor in a closed
container at
temperature T. Solid
line for the position
of the surface, and
the dashed line
shows the effect of
an infinitesimal
expansion of the
droplet.
Chap 18. Molecular interactions
167
Surface Energy of droplet
• The surface area AS is related to the volume VL of the liquid
and to the amount nL of the liquid.
4 3
VL  π r
3
AS  4 π r 2
dVL  4 π r 2dr
dAS  8 π r dr
• At constant V and T, the work of surface formation is equal
to the change in Helmholtz energy (dA) of the system.
• The fundamental equations at constant T for two phases are:
2
dAL T  -PL dVL  μL dnL  γ dAS
dAS  dVL
r
dAV T  -PV dVV  μV dnV
• At equilibrium:
dA T
 0  PV -PL  dVL 
2γ
dVL
or PL-PV  
2γ
r
r
where dA = dAL + dAV ; dVV = - dVL ; dnL = - dnV ; V =  L ;
Chap 18. Molecular interactions
Phase Equilibrium
168
• The Laplace equation for the vapor pressure at a
curved surface show that, as bubble or cavity becomes
larger, the pressure difference across its surface
decreases.
Pin - Pout = 2 γ/ r
where γ is the surface tension and r is the radius,
Thus the pressure inside the curvature is higher than
the pressure outside the boundary, Pin > Pout .
Chap 18. Molecular interactions
Surface Energy of cavity
169
• For a cavity, the outward force = Pressure x area = 4πr 2Pin
where the pressure inside is Pin and its radius is r.
• The inward force from the external pressure = 4 π r 2Pout
• The change in surface area when the radius of sphere change
from r to r + dr is
2
dAS  4 π r  dr  - 4 π r 2  8 π r dr
• As work = force x distance. The work done when the surface
is stretched by dr is γdAS
dw  F x dr  8 πγr dr
• The force opposing stretching through a distance dr when the
radius is r. At equilibrium, the outward and inward forces are
balanced. The force inwards arises from both external pressure
and the surface tension:4 π r 2Pin = 4 π r 2Pout + 8 π γr
• Now we have
Pin - Pout = 2 γ/ r
Chap 18. Molecular interactions
170
Chap 18. Molecular interactions
171
The balance of forces that results in
a contact angle, θc.
γsg = γsl + γlg cos θc.
Chap 18. Molecular interactions
Capillary rise and surface tension
172
• Capillary action is the tendency of liquids to rise up
capillary tubes of narrow bore. The height h to which
a liquid of mass density ρ and surface tension γ will
rise is
2γ
h
ρ gr
where g is the acceleration of free fall.
• There may be a nonzero angle between the edge of
the meniscus and the wall called the contact angle θc,
• If the contact angle between the tube and the liquid is
θc rather than zero, then the right hand side will
multiply by cos(θc).
1
γ  ρ g hr cosC 
2
Chap 18. Molecular interactions
Illustration
Illustration
• If water at 25℃ rises through 7.36 cm in a capillary of
radius 0.20 mm, its surface tension at that
temperature is
1
γ  ρ g hr
2
= ½ x(997.1 kg m-3)x(9.81 m s-2)x(7.36x10-2
m)x(2.0x10-4 m)
=72 mN m-1
where we have used 1 kg m s-2 = 1 N
Chap 18. Molecular interactions
173
174
Chap 18. Molecular interactions
Phase Equilibrium
175
• 溫度升高，表面張力下降，當達到臨界溫度TC 時，表面張力
趨向於零。這可用熱力學公式說明：
dG  S dT V dP   dA   B dnB
運用全微分的性質，可得：
B
S

( )T ,P ,nB  (
) A ,P ,nB
A
T
等式左方為正值，因為表面積增加，熵總是增加的。所以 γ
隨T 的增加而下降。
Chap 18. Molecular interactions
176
S

( )T ,P ,nB  (
) A ,P ,nB
A
T
Chap 18. Molecular interactions
Contact Angle and Interfacial Tension
177
• The contact angle θc , is related to the surface tensions of the
solid/gas; solid/liquid; and liquid/gas interfacial surface tension:
γsg; γsl; γlg; by
cos θc = (γsg- γsl) / γlg ;
• Capillary rise occurs for 0° < θc< 90°
• Capillary depress occurs for 90° < θc< 180°
An insect walking on the water
Chap 18. Molecular interactions
178
= 0°
-
< 90°
>0
= 90°
> 90°
=0
Chap 18. Molecular interactions
= 180°
<0
179
• Contact angle
• γ sg = γ sl + γ lg cos θc
• cos θc = (γ sg - γ sl ) / γ lg
• If we note that the superficial work of adhesion of the
liquid to the solid (the work of adhesion divided by
the area of contact) is wad, then
• wad = γ sg + γ lg - γ sl
• cos θc = (wad - γ lg )/ γ lg = (wad / γ lg) -1
Chap 18. Molecular interactions
180
cos θc = (wad /γlg ) -1
not wets
wets
Chap 18. Molecular interactions
Surface Wetting
• The criteria for surface wetting is θc >0°
corresponds to the wetting of a surface, and it is
possible when wad > 2 γlg
• equation can be written as:
cos C  
Wad
 lg
-1
Chap 18. Molecular interactions
181
Surface Wetting
• the liquid 'wets' (spreads over) the surface,
corresponding to 0 < θc < 90°, when 1 < wad /γlg < 2
• The liquid does not wet the surface, corresponding to
90° < θc < 180°, when 0 < wad /γlg < 1.
• For mercury in contact with glass, θc =140°, which
corresponds to wad /γlg =0.23, indicating a relatively
low work of adhesion of the mercury to glass on
account of the strong cohesive forces within mercury.
Chap 18. Molecular interactions
182
Surface Wetting
• The criteria for surface wetting is θc >0°
corresponds to the wetting of a surface, and it is
possible when wad > 2 γlg
(Source: Fonds der Chemischen Industrie, Germany; imageseries "Tenside")
Chap 18. Molecular interactions
183
18.8 Condensation
184
• The vapor pressure of a liquid depends on the
pressure applied to the liquid. Because curving a
surface gives rise to a pressure differential of 2 γ /r,
we can expect the vapor pressure above a curved
surface to be different from that above a flat surface.
By substituting this value of the pressure difference
into eqn 4.3 (P = P * exp(2γV̿ L /rRT) here P is the
vapor pressure when the pressure difference is zero)
we obtain the Kelvin equation for the vapor pressure
of a liquid when it is dispersed as droplets of radius r:
• P = P * exp(2γV̿ L /rRT)
Chap 18. Molecular interactions
Chap 18 Phase Equilibrium
• For droplet in a vapor:
For cavity in a liquid:
2γ
2γ
PV -PL 
PL-PV 
r
r
• When the radius of curvature is increased to infinity,
this pressure difference disappears, as it must for a
planar surface.
• The pressure is always greater on the concave side(凹
面) of the surface.
• The volume of droplet VL equals to the molar volume
V̿L times the amount nL :
 2
 2
dAS    dVL    V L dnL
r 
r 
Chap 18. Molecular interactions
185
186
Equilibrium Vapor Pressure
• The fundamental equation for G for the liquid phase is
dG L T,P  μ *L dnL  2/r V L dnL


 μL*  2V L/r dnL
 μL dnL
• The fundamental equation for G for the vapor phase is
dG V T,P  μ V dn V
• At equilibrium
dG T,P  0  dG L T,P  dG V T,P  μL*  2V L/r dnL  μ V dn V

dnL  -dn V ; μ V  μL*  2V L/r  μL
Chap 18. Molecular interactions

Kelvin equation
187
• Assuming that the vapor is an ideal gas,
μV   RT lnP/P   μL*  2 γV L /r
• For a flat surface, (r →∞)
μV   RT lnP * /P   μL*
• The difference gives the Kelvin equation:
RT lnP/P *   2 γV L /r
• For the vapor pressure of liquid that is dispersed as droplet
of radius r , P = P * exp(2γV̿ L /rRT)
• P* is the vapor pressure of the bulk liquid of molar volume ̿V̿
and γ is the surface tension of the liquid.
• For the vapor pressure inside a spherical cavity of radius r
change the sign in the exponent: P = P * exp(-2γV̿ V /rRT)
Chap 18. Molecular interactions
L
188
Fig 6.7
Fig 6.7 Effect of radius of curvature of a surface on the
vapor pressure of water at 25 ℃ . (a) for droplet and
(b) for cavity.
Droplet
(a)
(b)
 2V in
P P *  exp 
 rRT
Bubble
Chap 18. Molecular interactions




189
• The analogous expression for the vapor pressure inside
a cavity can be written at once. The pressure of the
liquid outside the cavity is less than the pressure inside,
so the only change is in the sign of the exponent in the
last expression.
• For droplets of water of radius I μm and I nm the ratios
p/p* at 25℃ are about 1.001 and 3, respectively.
• The second figure, although quite large, is unreliable
because at that radius the droplet is less than about 10
molecules in diameter and the basis of the calculation is
suspect. The first figure shows that the effect is usually
small; nevertheless it may have important
consequences.
Chap 18. Molecular interactions
190
Phase Equilibrium
Example The size of a water droplet when condensation starts
Water vapor is rapidly cooled to 25 ℃ to find the degree of
supersaturation required to nucleate water droplets
spontaneously. It is found that the vapor pressure of water must
be four times its equilibrium vapor pressure at 25 ℃.
(a) Calculate the radius of a stable water droplet formed at this
degree of supersaturation. (b) How many water molecules are
there in the droplet?
Ans: Assumed that the surface tension is independent of the
radius of curvature,
2V L
2 0.07197 N m-1  18x10 -6 m3 mol -1 
a)r 


0.75
nm
8.3145 J K -1 mol -1 298 K  ln 4
RT ln P/P sat 
3
4
4
-9
3
πr ρ
 0.75x10 m 1x10 3 kg m -3
3

b) N  3
M/N A
18x10 -3 kg mol -1 6.022x10 23 mol




Chap 18. Molecular interactions

-1

 59
Phase Equilibrium
191
• A metastable phase (亞穩定相): A thermodynamically unstable phase but
prevented by kinetic reason from a transition to a stable phase.
• A superheated liquid is a liquid above its boiling temperature.
• A supercooled liquid is a liquid below its freezing point.
• When a vapor is cooled such that its chemical potential lies above the
chemical potential of its liquid. This thermodynamically unstable vapor
phase is said to be supersaturated (過飽和).
• It is only unstable with respect to the bulk liquid but not small droplets.
• The nucleate (起成核作用) means to provide a center at which a
condensation process can occur at spontaneous nucleation centers (自發凝
結核核中心 ).
• This is the chance formation of droplets that are large enough to initiate
condensation in a supersaturated phase.
人工造霧
機:
Chap 18. Molecular interactions
192
• Consider the formation of a cloud: Warm, moist air rises
into the cooler regions higher in the atmosphere. At
some altitude the temperature is so low that the vapor
becomes thermodynamically unstable with respect to
the liquid and we expect it to condense into a cloud of
liquid droplets.
• The initial step can be imagined as a swarm of water
molecules congregating into a microscopic droplet.
Because the initial droplet is so small it has an enhanced
vapor pressure. Therefore, instead of growing it
evaporates. This effect stabilizes the vapor because an
initial tendency to condense is overcome by a
heightened tendency to evaporate.
Chap 18. Molecular interactions
193
• The vapor phase is then said to be supersaturated. It
is thermodynamically unstable with respect to the
liquid but not unstable with respect to the small
droplets that need to form before the bulk liquid
phase can appear, so the formation of the latter by a
simple, direct mechanism is hindered.
Chap 18. Molecular interactions
194
• Clouds do form, so there must be a mechanism. Two
processes are responsible.
• The first is that a sufficiently large number of
molecules might congregate into a droplet so big that
the enhanced evaporative effect is unimportant. The
chance of one of these sponetaneous nucleation
centers forming is low, and in rain formation it is not
a dominant mechanism.
• The more important process depends on the presence
of minute dust particles or other kinds of foreign
matter. These nucleate the condensation (that is,
provide centers at which it can occur) by providing
surfaces to which the water molecules can attach.
Chap 18. Molecular interactions
195
• Liquids may be superheated above their boiling
temperatures and supercooled below their freezing
temperatures. In each case the thermodynamically stable
phase is not achieved on account of the kinetic
stabilization that occurs in the absence of nucleation
centers. For example, superheating occurs because the
vapor pressure inside a cavity is artificially low, so any
cavity that does form tends to collapse. This instability is
encountered when an unstirred beaker of water is heated,
for its temperature may be raised above its boiling point.
Violent bumping often ensues as spontaneous nucleation
leads to bubbles big enough to survive. To ensure
smooth boiling at the true boiling temperature,
nucleation centers, such as small pieces of sharp-edged
glass or bubbles (cavities) of air, should be introduced.
Chap 18. Molecular interactions
雲霧的成因
• 霧和雲 是由浮游在空中的小水滴或冰晶組成的水汽凝結物，只是霧生
成在大氣的近地面層中，而雲生成在大氣的較高層而已。霧既然是水汽
凝結物，因此應從造成水汽凝結的條件中尋找它的成因。大氣中水汽達
到飽和的原因不外兩個：一是由於蒸發，增加了大氣中的水汽；另一是
由於空氣自身的冷卻。對於霧來說冷卻更重要。當空氣中有凝結核時，
飽和空氣如繼續有水汽增加或繼續冷卻，便會發生凝結。凝結的水滴如
使水準能見度降低到1千米以內時，霧就形成了。
• 另外，過大的風速和強烈的擾動不利於霧的生成。
• 因此，凡是在有利於空氣低層冷卻的地區，如果水汽充分，風力微和，
大氣層結穩定，並有大量的凝結核存在，便最容易生成霧。一般在工業
區和城市中心形成霧的機會更多，因為那裏有豐富的凝結核存在。
Chap 18. Molecular interactions
196
Further Information 18.2
• Crossed beam technique
• When an incident molecular beam is directed in to a
second molecular beam in order to study the
collisions of these highly selected states, the method
is called a crossed beam technique.
Chap 18. Molecular interactions
197
198
• Incident beam flux
• The incident beam flux I is the number of particles
from the incident molecular beam passing through a
given target area.
• Differential scattering cross-section
• In molecular beam work, the differential scattering
cross-section, σ, is the constant of proportionality
between the number of molecules that are scattered
per unit time into a cone that represents the area
covered by the eye of the detector, dI , and the
intensity, I, of the incident beam, the number density
of target molecules, and the path length, dx, through
the sample: dI = σ I N dx
Chap 18. Molecular interactions
199
Chap 18. Molecular interactions
200
Chap 18. Molecular interactions
201
Chap 18. Molecular interactions
state-to-state cross section
202
• A more detailed version of this quantity is the stateto-state cross section σnn’ , for scattering from an
initial state n to a final state n’, which depends among
other things, on the speed of approach υ.
Chap 18. Molecular interactions
203
Chap 18. Molecular interactions
204
Chap 18. Molecular interactions
205
Chap 18. Molecular interactions
206
Chap 18. Molecular interactions
207
Chap 18. Molecular interactions
208
Chap 18. Molecular interactions
homopolar contribution
209
• homopolar contribution (同極的貢獻 )
One of the contribution to the dipole moment of a
molecule is the imbalance in atomic radii result in an
imbalance of charge density because the enhanced
charge density associated with the overlap region
close to the nucleus of smaller atom.
• The diagram shows how the charge accumulation
leads to a region of negative charge closer to the
smaller atom.
• Such a "homopolar dipole" contribution to the total
dipole moment can arise even in the absence of a
difference in electronegativity between the two atoms.
Chap 18. Molecular interactions
210
18.1 One of the
contributions to the
dipole moment of a
molecule is the
imbalance of charge
arising from the overlap
of orbitals of different
radii. This diagram
shows how the charge
accumulation leads to a
region of negative
charge closer to the
smaller atom.
Chap 18. Molecular interactions
211
18.27 The radial distribution function of the oxygen atoms in liquid
water at three temperatures. Note the expansion as the
temperature is raised.
Source: A.H. Narten, M.D. Danford, and H.A. Levy, Discuss. Faraday
Soc. 43, 97 (1967)
Chap 18. Molecular interactions
212
18.28 The radial distribution
function for a simulation of
a liquid using impenetrable
hard spheres (ball-bearings)
Chap 18. Molecular interactions